Optimal. Leaf size=87 \[ \frac{2 x \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{n+2}{2 (j-n)};\frac{n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
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Rubi [A] time = 0.0525132, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2011, 365, 364} \[ \frac{2 x \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{n+2}{2 (j-n)};\frac{n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
Antiderivative was successfully verified.
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Rule 2011
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \sqrt{a x^j+b x^n} \, dx &=\frac{\left (x^{-n/2} \sqrt{a x^j+b x^n}\right ) \int x^{n/2} \sqrt{b+a x^{j-n}} \, dx}{\sqrt{b+a x^{j-n}}}\\ &=\frac{\left (x^{-n/2} \sqrt{a x^j+b x^n}\right ) \int x^{n/2} \sqrt{1+\frac{a x^{j-n}}{b}} \, dx}{\sqrt{1+\frac{a x^{j-n}}{b}}}\\ &=\frac{2 x \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{2+n}{2 (j-n)};1+\frac{2+n}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )}{(2+n) \sqrt{1+\frac{a x^{j-n}}{b}}}\\ \end{align*}
Mathematica [A] time = 0.142623, size = 134, normalized size = 1.54 \[ \frac{2 x \left (a (j-n) x^j \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2 j-n+2}{2 j-2 n};\frac{4 j-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )-(2 j-n+2) \left (a x^j+b x^n\right )\right )}{(n+2) (-2 j+n-2) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.387, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a{x}^{j}+b{x}^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x^{j} + b x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x^{j} + b x^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a x^{j} + b x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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